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To show why ∆G=0 at equilibrium from a mathematical perspective, use ∆G˚= - RTlnK and ∆G = ∆G˚ + RTlnQ. If we substitute ∆G˚ out, then ∆G = RTlnQ - RTlnK. When the reaction is at equilibrium, the Q=K, thus ∆G = RTlnQ - RTlnK = RTlnK - RTlnK = 0.
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